Am I obligated to slaughter a sheep to fulfill a vow, without specifying the animal, based on common custom?

Initiating a vow (nadhr) is absolutely disliked (makrooh). If the vow is conditional, then it is obligatory to fulfill it upon the occurrence of the condition. So, if you succeed in your studies, slaughtering becomes obligatory upon you. It suffices for you to perform one of the three things you intended. And slaughtering a sheep carries greater reward due to the greater benefit it provides.